Finding Mixed Strategy Nash Equilibrium for Continuous Games through Deep Learning

Nash equilibrium has long been a desired solution concept in multi-player games, especially for those on continuous strategy spaces, which have attracted a rapidly growing amount of interests due to advances in research applications such as the generative adversarial networks. Despite the fact that several deep learning based approaches are designed to obtain pure strategy Nash equilibrium, it is rather luxurious to assume the existence of such an equilibrium. In this paper, we present a new method to approximate mixed strategy Nash equilibria in multi-player continuous games, which always exist and include the pure ones as a special case. We remedy the pure strategy weakness by adopting the pushforward measure technique to represent a mixed strategy in continuous spaces. That allows us to generalize the Gradient-based Nikaido-Isoda (GNI) function to measure the distance between the players' joint strategy profile and a Nash equilibrium. Applying the gradient descent algorithm, our approach is shown to converge to a stationary Nash equilibrium under the convexity assumption on payoff functions, the same popular setting as in previous studies. In numerical experiments, our method consistently and significantly outperforms recent works on approximating Nash equilibrium for quadratic games, general blotto games, and GAMUT games.